The logarithmic-length conjecture for binary words with a prescribed number of subsequences

A binary word is a finite word over the alphabet $\{\mathsf{A},\mathsf{B}\}$. For a binary word $\mathfrak{w}$, let $\mathsf{P}(\mathfrak{w})$ denote its number of nonempty subsequences.

Logarithmic-length conjecture. For every integer $n\geqslant 1$, there exists a binary word $\mathfrak{w}$ such that

$$|\mathfrak{w}|=O(\log n)$$

and

$$\mathsf{P}(\mathfrak{w})=n.$$

The Fibonacci-type alternating words show that any binary word with exactly $n$ subsequences must have length $\Omega(\log n)$, so the conjecture asks whether this lower bound is sharp up to a constant factor. The source does not provide a resolution.